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叶鸿国 Hong-Gwa Yeh 中央大学 , 台湾 hgyeh@math.ncu.tw July 31, 2009

Some Results on Labeling Graphs with a Condition at Distance Two. 叶鸿国 Hong-Gwa Yeh 中央大学 , 台湾 hgyeh@math.ncu.edu.tw July 31, 2009. Channel-Assignment Problem. Hale, 1980. Hale, 1980, IEEE. 1. 1. 1. 1. 2. 1. 1. 2. 2. 2. 3. 1. 3. 1. 1. 3. 1. Chromatic number = 3. 2. 2.

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叶鸿国 Hong-Gwa Yeh 中央大学 , 台湾 hgyeh@math.ncu.tw July 31, 2009

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  1. Some Results on Labeling Graphs with a Condition at Distance Two 叶鸿国Hong-Gwa Yeh 中央大学,台湾 hgyeh@math.ncu.edu.tw July 31, 2009

  2. Channel-Assignment Problem

  3. Hale, 1980

  4. Hale, 1980, IEEE

  5. 1 1

  6. 1 1

  7. 2 1

  8. 1 2 2 2 3 1 3 1 1 3

  9. 1 Chromatic number = 3 2 2 2 3 1 3 1 1 3

  10. However, interference phenomena may be so powerful that even the different channels used at “very close” transmitters may interfere.

  11. Roberts, 1988 ? “close” transmitters must receive different channels and “very close”transmitters must receive channels that are at least two channels apart. ?

  12. Griggs and Yeh, 1992, SIAM J. Discrete Math. k-L(2,1)-labeling of a graph G

  13. k-L(2,1)-labeling of a graph G f:V(G)-------->{0,1,2,…,k} s.t. |f(x)-f(y)|≧2 if d(x,y)=1 |f(x)-f(y)|≧1 if d(x,y)=2 J. R. GRIGGS R. K. YEH

  14. 1 Roberts, 1980 2 2 2 3 1 3 1 1 3

  15. 8-L(2,1)-labeling of P 7-L(2,1)-labeling of P 6-L(2,1)-labeling of P ? ? ?

  16. 8 3 9-L(2,1)-labeling of P ?

  17. 8 3 9-L(2,1)-labeling of P ? λ(G) =λ-number of G λ(P)=9

  18. The problem of determining λ(G) for general graphsG is known to be NP-complete!

  19. Goodupper boundsfor λ(G) are clearly welcome.

  20. Griggs and Yeh: λ(G) ≦△2+2△ Chang and Kuo:λ(G) ≦△2+△ Kral and Skrekovski : λ(G) ≦△2+△-1 Goncalves:λ(G) ≦△2+ △-2

  21. Griggs-Yeh Conjecture 1992 J. R. GRIGGS λ(G) ≦△2 for any graph G with maximum degree △≧2 R. K. YEH

  22. Very recently Havet, Reed, and Sereni have shown that Griggs-Yeh Conjecture holds for sufficiently large△ !! SODA 2008

  23. Note that to prove Griggs-Yeh Conjecture it suffices to consider regular graphs.

  24. However….

  25. Very little was known about exact L(2,1)-labeling numbers for specific classes of graphs. --- even for3-regular graphs

  26. Consider various subclasses of 3-regular graphs Kang, 2008, SIAM J. on Discrete Math., proved that Griggs-Yeh Conjecture is true for 3-regular Hamiltoniangraphs

  27. Other important subclasses of 3-regular graphs Generalized Petersen Graph

  28. GeneralizedPetersenGraph of order 5 GPG(5)

  29. GPG(3) , GPG(4)

  30. GPG(6)

  31. GPG(9)

  32. Griggs-Yeh Conjecture says that λ(G) ≦9for all GPGs G

  33. Georges and Mauro, 2002, Discrete Math. λ(G) ≦8for all GPGs G except for the Petersen graph

  34. Georges and Mauro, 2002, Discrete Math. λ(G) ≦7for all GPGs G of order n≦6 except for the Petersen graph

  35. Georges-Mauro Conjecture 2002 For any GPG G of order n≧7, λ(G) ≦7

  36. Jonathan Cass Denise Sakai Troxell Sarah Spence Adams 2006, IEEE Trans. Circuits & Systems Georges-Mauro Conjecture is true for orders 7 and 8

  37. More….

  38. Number of non-isomorphic GPGs of order n with the aid of a computer program

  39. Y-Z Huang, C-Y Chiang, L-H Huang, H-G Yeh 2009 Georges-Mauro Conjecture is true for orders 9,10,11 and 12

  40. Generalized Petersen graphs of orders 9, 10, 11 and 12 Theorem One-page proof !! 42

  41. 3 3 3 43

  42. 3 1, 2, 4, 5, 6 3 3 44

  43. Case 7 Case 3 Case 5 Case 6 Case 2 Case 1 Case 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 45

  44. Case 1 3 5 1 6 2 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 46

  45. Case 1 3 Case A 5 1 6 2 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 47

  46. Case 1 3 Case A 0 7 5 1 6 2 7 0 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 0 7 48

  47. Case 1 3 Case B 5 1 6 2 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 49

  48. Case 1 3 Case B 7 0 5 1 6 2 0 7 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 7 0 50

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